test.pi

-- Leibniz equality
Eq : [A : Set] -> (x y : A) -> Set
   = \ A x y -> (P : A -> Set) -> P x -> P y

refl : [A : Set] -> [x : A] -> Eq x x
     = \ P px -> px

sym' : [A : Set] -> (x y : A) -> Eq x y -> Eq y x
     = \ x y eqxy P py -> eqxy (\z -> P z -> P x) (\px -> px) py

sym : [A : Set] -> [x y : A] -> Eq x y -> Eq y x = sym' _ _

foo : _ = pair

-- Categories
Cat : Set
    = (Obj   : Set)
    * (Arr   : Obj -> Obj -> Set)
    * (id    : (A : Obj) -> Arr A A)
    * (comp  : (A B C : Obj) -> Arr B C -> Arr A B -> Arr A C)
    * (idl   : (A B : Obj) -> (f : Arr A B) -> Eq (comp A B B (id B) f) f)
    * (idr   : (A B : Obj) -> (f : Arr A B) -> Eq (comp A A B f (id A)) f)
    * (assoc : (A B C D : Obj) -> (f : Arr C D) -> (g : Arr B C) -> (h : Arr A B) ->
               Eq (comp A B D (comp B C D f g) h)
                  (comp A C D f (comp A B C g h)))
    * One

-- Projections
Obj : Cat -> Set = fst
Arr : (Z : Cat) -> Obj Z -> Obj Z -> Set
  = \Z -> fst (snd Z)
id  : (Z : Cat) -> [A : Obj Z] -> Arr Z A A
  = \Z -> fst (snd (snd Z)) _
comp : (Z : Cat) -> [A B C : Obj Z] -> Arr Z B C -> Arr Z A B -> Arr Z A C
  = \Z -> fst (snd (snd (snd Z))) _ _ _
idl : (Z : Cat) -> [A B : Obj Z] -> (f : Arr Z A B) -> Eq (comp Z (id Z) f) f
  = \Z -> fst (snd (snd (snd (snd Z)))) _ _
idr   : (Z : Cat) -> [A B : Obj Z] -> (f : Arr Z A B) -> Eq (comp Z f (id Z)) f
  = \Z -> fst (snd (snd (snd (snd (snd Z))))) _ _
assoc : (Z : Cat) -> [A B C D : Obj Z] -> (f : Arr Z C D) -> (g : Arr Z B C) -> (h : Arr Z A B) ->
        Eq (comp Z (comp Z f g) h)
           (comp Z f (comp Z g h))
  = \Z -> fst (snd (snd (snd (snd (snd (snd Z)))))) _ _ _ _